connected dominating set in sensor networks and...
TRANSCRIPT
Handbook of Combinatorial OptimizationD.-Z. Du and P. Pardalos (Eds.) pp. 329 - 369
c©2004 Kluwer Academic Publishers
Connected Dominating Set in Sensor Networksand MANETs
Jeremy Blum Min Ding Andrew Thaeler Xiuzhen ChengDepartment of Computer ScienceThe George Washington University, Washington, DC 20002E-mail: blumj,minding,athaeler,[email protected]
Contents
1 Introduction 330
2 Network Model and CDS Applications 3322.1 Network Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3322.2 Applications of CDS in Wireless Networks . . . . . . . . . . . . . . . 334
3 Centralized CDS Construction 3353.1 Guha and Khuller’s Algorithm . . . . . . . . . . . . . . . . . . . . . 3363.2 Ruan’s Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3383.3 Cheng’s Greedy Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 3403.4 Min’s Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3413.5 Butenko’s Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 341
4 Distributed CDS Construction 3414.1 WCDS Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . 3424.2 Greedy CDS Construction . . . . . . . . . . . . . . . . . . . . . . . . 3444.3 MIS Based CDS Construction . . . . . . . . . . . . . . . . . . . . . . 345
4.3.1 Alzoubi and Wan’s Single Leader Algorithm . . . . . . . . . . 3454.3.2 Cheng’s Single Leader Algorithm . . . . . . . . . . . . . . . . 3474.3.3 Alzoubi’s Multiple Leader algorithm . . . . . . . . . . . . . . 3494.3.4 Cheng’s Multiple Leader Algorithm . . . . . . . . . . . . . . 3504.3.5 Single Leader vs. Multiple Leader . . . . . . . . . . . . . . . 350
4.4 Pruning Based CDS Construction . . . . . . . . . . . . . . . . . . . . 3514.4.1 Wu and Li’s Algorithm . . . . . . . . . . . . . . . . . . . . . 3514.4.2 Butenko’s algorithm . . . . . . . . . . . . . . . . . . . . . . . 353
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4.5 Multipoint Relaying Based CDS Construction . . . . . . . . . . . . . 3534.6 Steiner Tree Based CDS Construction . . . . . . . . . . . . . . . . . 3544.7 Proactively Considering Nodal Mobility in CDS Construction . . . . 354
5 Analysis of Distributed CDS Algorithms 3565.1 A Toolbox for CDS Construction . . . . . . . . . . . . . . . . . . . . 357
5.1.1 WCDS Construction Techniques . . . . . . . . . . . . . . . . 3575.1.2 CDS Construction Techniques . . . . . . . . . . . . . . . . . . 357
5.2 Selection of CDS Construction Methods . . . . . . . . . . . . . . . . 360
6 Conclusions and Discussions 361
References
1 Introduction
Recently developed classes of wireless networks have blurred the distinctionbetween the network infrastructure and network clients. Sensor networks,for example, consist of one or more base stations and a large number ofinexpensive nodes, which combine sensors and low power wireless radios.Due to limited radio range and battery power, most nodes cannot commu-nicate directly with a base station, but rather rely on their peers to forwardmessages to and from base stations. Likewise, in mobile ad hoc networks(MANETs), the routing of messages is also performed by ordinary nodes. Infact, a MANET typically has no network infrastructure, therefore all routingand network management functions must be performed by ordinary nodes.
The key to scalability and efficiency in traditional computer networks isthe organization of the network infrastructure into a hierarchical structure.However, due to the lack of a network infrastructure, sensor networks andMANETs are inherently flat. In order to achieve scalability and efficiency,new algorithms have emerged that rely on a virtual network infrastruc-ture, which organizes ordinary nodes into a hierarchy. The construction ofthis infrastructure is the primary application of Connected Dominating Sets(CDSs) in wireless networks.
The utility of CDSs in wireless ad hoc networks has been demonstrated inprotocols that perform a wide range of communication functions. CDSs haveformed an underlying architecture used by protocols including media accesscoordination [8, 43, 64]; unicast [35, 33, 34, 76], multicast/broadcast [51,52, 67, 73, 78, 79, 80], and location-based routing [36]; energy conservation
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[19, 38, 63, 72, 82]; and topology control [37, 38]. CDS can also be used tofacilitate resource discovery in MANET [45, 48].
In this chapter, we are going to survey the CDS construction techniquesproposed in the context of sensor networks and MANETs. Sections 3 and 4address details of centralized and distributed algorithms respectively. The-oretically any centralized algorithm can be implemented in a distributedfashion, with the tradeoff of higher protocol overhead. We are going toexamine several centralized algorithms and their corresponding distributedimplementations in detail. Distributed or even localized algorithms are veryimportant for sensor networks and MANETs. CDS must be constructedefficiently to be applicable in a mobile or large scale network. Due to thedynamism of wireless links and nodal mobility, algorithms should rely onlimited knowledge of the current network topology.
Note that the design goals of different algorithms vary based on the needsof the protocols making use of the CDS. When designing a CDS algorithm,one must take the following parameters into consideration: performancebounds, degree of localization, time and message complexities, and stabilitywith respect to nodal movement. We are going to analyze these algorithmsin Section 5. We will provide a toolbox of techniques (Subsection 5.1) elicitedfrom the examination of CDS construction algorithms, and present guide-lines (Subsection 5.2) to aid in the selection of particular techniques basedon the design goals of an application.
Actually many works seek a minimum connected dominating set (MCDS)in unit-disk graphs as their major design goal. Thus performance boundsis their primary design parameter. The rationale of this problem formula-tion can be justified as follows. The foot print of an ad hoc network withfixed transmission range for each host can be modelled by a unit-disk graph[25]. And minimizing the cardinality of the computed CDS can help to de-crease the control overhead since broadcasting for route discovery [47, 60]and topology update [30] is restricted to a small subset of nodes [25]. There-fore broadcast storm problem [59] inherent to global flooding can be greatlydecreased.
Other works seek a connected dominating set that provides good resourceconservation property [19, 76]. Thus performance bound is not their primaryconsideration. Instead, the hop count of communication path between nodesis taken into consideration for load balance [19] and power conservation[78, 79]. In the following section (Section 2), we will present network modeland useful definitions needed for algorithm elaboration. We also will givean overview of CDS applications.
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2 Network Model and CDS Applications
The following section provides background information for the analysis ofCDS applications in ad hoc wireless networks. We first present a math-ematical model for the networks under consideration and introduce usefulterminologies and definitions from graph theory. Then we sketch the variouswireless network applications utilizing connected dominating sets.
2.1 Network Model
An ad hoc wireless network can be represented by a graph G(V, Et) com-prised of a set of vertices V and time-varying edges Et. For each pair ofvertices u, v ∈ V , (u, v) ∈ Et if and only if the nodes u and v are withincommunication range. Due to nodal movement, the topology of the networkis dynamic, as reflected by Et.
Given omni-directional antennae, the communication range of a node ina wireless network is typically modelled as a disk centered at the node withradius equal to the transmission range of the radio. Consequently, whentransmission range is fixed for all nodes, the network has the property of aunit-disk graph (UDG), where an edge exists if and only if two nodes haveinter-nodal distance less than or equal to 1 unit ( the fixed communicationrange). Many of the CDS algorithms use the properties of UDG’s to provetheir performance bounds.
Each node v has an associated set of nodal properties. Typical propertiesinclude the following:
• IDv, the unique ID for node v.
• loctv, the location of node v at time t.
• velocitytv, the velocity vector for node v at time t.
A number of definitions from graph theory are used in this chapter.Figure 1 can help to illustrate the following concepts:
• Open Neighbor Set, N(u) = v | (u, v) ∈ E, is the set of nodes thatare neighbors of u. In Figure 1, the open neighbor set of e is d, f, g.
• Closed Neighbor Set, N [u] = N(u) ∪ u, is the set of neighbors of uand u itself. In Figure 1, the closed neighbor set of e is d, e, f, g.
• Maximum Degree, ∆, is the maximum count of edges emanating froma single node. The maximum degree of the graph in Figure 1 is three,and occurs at nodes c, e, and g.
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Figure 1: Representation of a wireless network with eight nodes as a graph.
• Independent Set, is a subset of V such that no two vertices within theset are adjacent in V . For example, a, b, f, h is an independent setin Figure 1.
• Maximal Independent Set (MIS), is an independent set such that addingany vertex not in the set breaks the independence property of the set.Thus, any vertex outside of the maximal independent set must be adja-cent to some node in the set. The previous independent set a, b, f, hmust have node d added to become an MIS.
• Dominating Set, S, is defined as a subset of V such that each nodein V − S is adjacent to at least one node in S. Thus, every MIS isa dominating set. However, since nodes in a dominating set may beadjacent to each other, not every dominating set is an MIS. Finding aminimum-sized dominating set or MDS is NP-Hard [41].
• Connected Dominating Set (CDS), C, is a dominating set of G whichinduces a connected subgraph of G. One approach to constructing aCDS is to find an MIS, and then add additional vertices as needed toconnect the nodes in the MIS. A CDS in Figure 1 is c, d, e, g.
• Minimum Connected Dominating Set (MCDS) is the CDS with min-imum cardinality. Given that finding minimum sized dominating setis NP-Hard, it should not be surprising that finding the MCDS is alsoNP-Hard [41]. In Figure 1, c, g is a minimum connected dominatingset.
• Weakly Connected Dominated Set (WCDS), S, is a dominating setsuch that N [S] induces a connected subgraph of G. In other words, the
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subgraph weakly induced by S is the graph induced by the vertex setcontaining S and its neighbors. Given a connected graph G, all of thedominating sets of G are weakly connected. Computing a minimumWCDS is NP-Hard [41].
• Steiner Tree, is a minimum weight tree connecting a given set of ver-tices in a weighted graph. After finding an MIS, connecting the nodestogether could be formulated as an instance of the Steiner Tree prob-lem. Like many of the other problems that arise in CDS construction,this problem is NP-Hard [41].
2.2 Applications of CDS in Wireless Networks
Sensor networks and MANETs have unique characteristics that require thedevelopment of protocols specific to them. For efficiency reasons, many ofthese protocols first organize the network through the construction of dom-inating sets. These protocols address media access, routing, power manage-ment, and topology control.
At the Link Layer, clustering can increase spatial reuse of the spectrum,minimize collisions, and provide Quality of Service (QoS) guarantees [8, 43,64, 65]. Correspondingly, the nodes in the dominating set can coordinatewith one another and use orthogonal spreading codes in their neighborhoodsto improve spatial reuse with code division spread spectrum techniques [42].Furthermore, these nodes can coordinate access to the wireless media bytheir neighbors for QoS or collision avoidance purposes.
As first noted by Ephremedis et al., a CDS can create a virtual net-work backbone for packet routing and control [40]. Messages can be routedfrom the source to a neighbor in the dominating set, along the CDS to thedominating set member closest to the destination node, and then finally tothe destination. This is termed dominating set based routing [33, 76], orBackbone based rouing [34], or spine based routing [35, 66]. Restricting therouting to the CDS results in a significant reduction in message overheadassociated with routing updates [18]. Furthermore, the dominating set canbe organized into a hierarchy to further reduce control message overhead[56, 64, 65].
A CDS is also useful for location-based routing. In location-based rout-ing, messages are forwarded based on the geographical coordinates of thehosts, rather than topological connectivity. Intermediate nodes are selectedbased on their proximity to the message’s destination. With this scheme, itis possible for a message to reach a local maximum, where it has been sent
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to an intermediate node whose neighbors are all further from the destinationthan itself. In this case, the routing must enter a recovery phase, where theroute may backtrack to find another path. However, if messages are onlyforwarded to nodes in the dominating set, the inefficiency associated withthis recovery phase can be greatly reduced [36].
The efficiency of multicast/broadcast routing can also be improved throughthe utilization of CDSs. A big problem in multicast/broadcast routing isthat many intermediate nodes unnecessarily forward a message. Nodes oftenhear the same message multiple times. This is the broadcast storm problem[59]. If the message is routed along a CDS, most of the redundant broadcastscan be eliminated [25, 51, 52, 67, 73, 78, 79, 80].
Nodes in a wireless network often have a limited energy supply. CDSsplay an important role in power management. They have been used toincrease the number of nodes that can be in an sleep mode, while stillpreserving the ability of the network to forward messages [19, 38, 82]. Theyhave also been used to balance the network management requirements toconserve energy among nodes [63, 72, 77, 78, 79, 80].
In large-scale dense sensor networks, sensor topology information extrac-tion can be handled by CDS construction [37, 38]. Other than routing, thevirtual backbone formed by dominating set can also be used to propagate“link quality” information for route selection for multimedia traffic [64], orto serve as database servers [49], etc.
3 Centralized CDS Construction
The first instance of a dominating set problem arose in the 1850’s, wellbefore the advent of wireless networks [70]. The objective of the five queensproblem is to find the minimum number of queens that can be placed on achessboard such that all squares are either attacked or occupied by a queen.This problem was formulated as a dominating set of a graph G(V ,E), withthe vertices corresponding to squares on the chessboard, and (u, v) ∈ E ifand only if a queen can move from the square corresponding to u to thesquare corresponding to v.
MCDS in general graphs was studied in [41], in which a reduction fromthe Set Cover Problem [41] to the MCDS problem was shown. This resultimplies that for any fixed 0 < ε < 1, no polynomial time algorithm withperformance ratio ≤ (1− ε)H(∆) exists unless NP ⊂ DTIME[nO(log log n)][54], where ∆ is the maximum degree of the input graph and H is theharmonic function. The MCDS remains NP-hard [29] for unit-disk graphs.
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MCDS in unit-disk graphs has constant performance ratio, as provedby [4, 16, 23, 69]. A polynomial time approximation scheme for computinga MCDS in unit-disk graphs has been developed by Cheng et al. in [26].A significant impact of this result is that a MCDS in unit-disk graphs canbe approximated to any degree if computing time is permitted. Note thatheuristics proposed for unit-disk graphs work well for general graphs, buttheir performance analysis is unapplicable. Thus in this section and next, wewill focus on algorithm description and skip the corresponding performanceanalysis. Note that we intentionally omit the fact that these algorithmsare proposed in either unit-disk graphs or general graphs because they areactually applicable in both graph models. Also note that we are going touse either the name of the first author or all authors’ names of the paper torepresent the algorithm.
In the following we will focus on centralized CDS construction algo-rithms. Distributed heuristics will be discussed in Section 4.
3.1 Guha and Khuller’s Algorithm
In 1998, Guha and Khuller proposed two CDS construction strategies intheir seminal work [44], which contains two greedy heuristic algorithms withbounded performance guarantees. In the first algorithm, the CDS is grownfrom one node outward. In the second algorithm, a WCDS is constructed,and then intermediate nodes are selected to create a CDS. The distributedimplementations of both algorithms were provided by Das et al. in [33],which will be addressed in Subsection 4.2. Many algorithms designed latter[23, 69] are motivated by either of these two heuristics. We sketch theprocedures in the following.
The first algorithm begins by marking all vertices white. Initially, thealgorithm selects the node with the maximal number of white neighbors.The selected vertex is marked black and its neighbors are marked gray. Thealgorithm then iteratively scans the gray nodes and their white neighbors,and selects the gray node or the pair of nodes (a gray node and one of itswhite neighbors), whichever has the maximal number of white neighbors.The selected node or the selected pair of nodes are marked black, with theirwhite neighbors marked gray. Once all of the vertices are marked grayor black, the algorithm terminates. All the black nodes form a connecteddominating set. This algorithm yields a CDS of size at most 2(1 + H(∆)) ·|OPT |, where H is the harmonic function, and OPT refers to an optimalsolution – that is, a minimum connected dominating set.
For example, consider the graph in Figure 2. Initially, either node c, or e,
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Figure 2: An example of Guha’s first algorithm.
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or g could be marked since they have maximal degree. Node g is arbitrarilypicked from these candidates and marked black. All its neighbors are thenmarked gray. We now consider the gray nodes e, f , h, and the pair of grayand white nodes (d, e). Of all these, the pair (d, e) covers the most number ofwhite neighbors - two. So we mark both d and e black. Finally, we considerthe gray node c and the pairs (c, a) and (c, b). All these candidates have thesame number of white neighbors. Therefore the single node c is selected.Now all the nodes are either black or gray, and the set of nodes in blackc, d, e, g forms a CDS.
The second algorithm also begins by coloring all nodes white. A pieceis defined to be either a connected black component, or a white node. Thealgorithm contains two phases. The first phase iteratively selects a node thatcauses the maximum reduction of the number of pieces. In other words, thegreedy choice for each step in the first phase is the node that can decreasethe maximum number of pieces. Once a node is selected, it is marked blackand its white neighbors are marked gray. The first phase terminates when nowhite node left. After the first phase, there exists at most |OPT | numberof connected black components. The second phase constructs a SteinerTree that connects all the black nodes by coloring chains of two gray nodesblack. The size of the resulting CDS formed by all black nodes is at most(3 + ln(∆)) · |OPT |.
Figure 3 shows an example of the second algorithm. First, node g ismarked as it is one of the nodes with the maximum number of white neigh-bors. Next, node c is marked because it can reduce the maximum numberof pieces compared with any other node. Now the first phase ends as thereis no white node left. In the second phase, a Steiner Tree is constructed byadding nodes d and e to connect nodes c and g.
3.2 Ruan’s Algorithm
The potential function used in the second algorithm of Guha and Khuller[44] is the number of pieces. Each step seeks maximum reduction in thenumber of pieces in the first phase. By modifying the potential function,Ruan et al. [62] proposes a one-step greedy approximation algorithm withperformance ratio at most 3+ ln(∆). This algorithm also requires each nodeto be colored white at the beginning. If there exists a white or gray nodesuch that coloring it black and its white neighbors gray would reduce thepotential function, then choose the one that causes maximum reduction inthe potential function.
The potential function plays a critical rule in this algorithm. It is defined
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Figure 3: An example of Guha’s second algorithm.
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in the following way in [62]. Given a connected graph G(V,E), define p(C)be the number of connected black components in the subgraph induced byC ⊂ V . Let D(C) be the set of all edges incident to vertices in C. Defineq(C) be the number of connected components in the subgraph G(V, D(C)).Then the potential function is defined to be f(C) = p(C) + q(C).
In the greedy algorithm, let C be the set containing all black nodes.Thus initially f(C) = |V | since C = φ. The first step chooses a node x withmaximum degree. Every other step selects a node x such that f(C)−f(C ∪x) is maximized. Color node x black and color all its white neighborsgray. The algorithm ends when f(C) = 2, where C is the resultant CDS.
3.3 Cheng’s Greedy Algorithm
In [24], Cheng et al. propose a greedy algorithm for MCDS in unit-diskgraphs. Compared to the many heuristics discussed in Subsection 4.3, thisalgorithm relies on an MIS but the resultant CDS may not contain all theelements in the MIS.
Assume initially all nodes are colored white. The construction of a CDScontains four phases. In the first phase, an MIS is computed and all itsmembers are colored red. In the second phase, a node that can decreasethe maximum number of pieces is selected, where a piece is either a rednode, or a connected black component. This node is colored black and allits non-black neighbors are colored gray. When the second phase is over, westill have some white nodes left. The third phase will compute a spanningtree for each connected component in the subgraph reduced by all whitenodes. Connect each tree to the nearest black component with black nodesaccordingly. All non-leaf tree nodes are colored black while leaf nodes arecolored gray. The last phase will seek chains of two gray nodes to connectdisjoint black components.
The motivation of Cheng’s algorithm are two fold. First, the greedychoice in Guha and Khuller’s second algorithm [44] is the one that can de-crease the maximum number of pieces, where a piece is either a connectedblack component, or a white node. Second, a unit-disk graph has at most5 independent neighbors. Thus intuitively one can choose the greedy choicethat can connect to as many independent nodes as possible. In other words,the node to be colored black at each step will try to cover more uncoveredarea, if we model vertices in a unit-disk graph as nodes in a flat area. Un-fortunately Cheng’s algorithm does not have a solid performance analysis.
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3.4 Min’s Algorithm
Recently Min et al. [58] propose to use a Steiner tree with minimum numberof Steiner nodes (ST-MSN) [20, 39, 53] to connect a maximal independentset. This algorithm contains two phases. The first phase constructs anMIS with the following property: every subset of the MIS is two hops awayfrom its complement. Color all nodes in the MIS black; color all othernodes gray. In the second phase, a gray node that is adjacent to at leastthree connected black components is colored black in each step. If no nodesatisfying this condition can be found, a gray node that is adjacent to at leasttwo connected black components will be colored black. This algorithm hasperformance ratio 6.8 for unit-disk graphs. Its distributed implementationis sketched in Subsection 4.6.
ST-MSN in Euclidean plane is NP-hard [53]. A 3-approximation algo-rithm for ST-MSN in Euclidean plane is proposed in [20] and is extendedto unit-disk graphs by Min et al. in [58]. Since the size of any MIS is atmost 3.8 · |OPT | + 1.2 [81] in unit-disk graphs, where OPT is any MCDS,the computed CDS by Min’s algorithm has size at most 6.8 ·OPT [58].
3.5 Butenko’s Algorithm
The heuristic proposed in [14, 15] is pruning-based. In other words, theconnected dominating set S is initialized to the vertex set of graph G(V, E),and each node will be examined to determine whether it should be removedor retained. Assume all nodes in S are colored white at the beginning.Define the effective degree of a node to be its white neighbors in S. Considera white node x ∈ S with minimum effective degree. If removing x from Smakes the induced graph of S disconnected, then retain x and color it black.Otherwise, remove x from S. At the same time, if x does not have a blackneighbor in S, color its neighbor with maximum effective degree in S black.Repeat this procedure until no white node left in S. This algorithm hastime complexity O(|V | · |E|). It does not have a performance analysis. Wewill discuss its distributed implementation in Subsection 4.4.2.
4 Distributed CDS Construction
For sensor networks and MANETs, distributed CDS construction is moreeffective due to the lack of a centralized administration. On the other hand,the large problem size (e.g. a sensor network may contain hundreds of thou-sands of sensors) also prohibits the centralized CDS computation. In this
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section, we survey a variety of distributed approaches that seek to balancethe competing requirements of complexity, running time, stability, and over-head. Note that many published results contain algorithms that differ verylittle from each other. To be more focus, we decide only to cover the majordistributed CDS construction techniques.
In wireless networks, CDS problems have been formulated in a num-ber of ways, depending on the needs of the particular application. Theseformulations can be classified into WCDSs, non-localized CDSs, localizedCDSs, and stable CDSs with nodal mobility. In this chapter, we chooseto use a different classification containing the following categories based onthe CDS construction techniques: WCDS, greedy CDS, MIS based CDS,pruning based CDS, multipoint forwarding based CDS, Steiner tree basedCDS, and stable CDS. In the following, we will examine each category andexplore example algorithms in detail. We also will sketch the maintenanceof the computed CDS if available.
4.1 WCDS Construction
In a WCDS, the vertex set is partitioned into a set of clusterheads andcluster members, such that each cluster member is within radio range of atleast one clusterhead. There are two ways of approaching this problem. Ifnodal location information is known, then the nodes can be clustered ge-ographically. Otherwise, nodes can be clustered based solely on the graphtopology. Chen and Liestman [21] propose a series of approximate algo-rithms for computing a small WCDS to be used to cluster mobile ad hocnetworks.
Geographical clustering algorithms create a virtual grid in the geograph-ical region where the network exists. Each cluster comprises all of the nodesin a given grid. The Grid Location Service, for example, uses this gridstructure to disseminate the location information of nodes throughout thenetwork. Using the grid structure, the density of nodes holding the locationinformation of other nodes decreases as the distance from the node increases[50]. Geographical Adaptive Fidelity attempts to minimize energy consump-tion in a network by clustering the nodes based on a grid, and having onlyone node per grid responsible for routing at any given time [82].
In the first WCDS algorithm for wireless networks, nodes elect theirneighbor with the lowest ID as their clusterhead [40]. Nodes learn aboutthe ID’s of their current neighbors through periodic beacons that each nodebroadcasts. Whenever a node with a lower ID moves into range of a clus-terhead, it becomes the new clusterhead.
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Highest Connectivity Clustering presents an improvement over LowestID Clustering by reducing the number of clusters [43]. In addition to pe-riodically broadcasting its ID, each node also broadcasts the size of N(u),its open neighbor set. A node becomes clusterhead if it is the most highlyconnected among its neighbors. When nodes are moving, this selection cri-terion is less stable than Lowest ID Clustering, since ID’s do not change,but nodal degree changes often.
One potential problem with Lowest ID Clustering is its unfairness. Nodesin the dominating set may be burdened with additional responsibilities. Tobe more equitable in clusterhead selection, the Min-Max D-Cluster algo-rithm selects nodes based on ID, but attempts to elect both low and highID nodes [7]. Another interesting feature of the algorithm is the creation ofd-hop dominating sets where each node is at most d hops from a node inthe dominating set.
A serious problem with the Lowest ID Clustering is that clusterheadselection can be very unstable. The left side of Figure 4 displays a clusterheaded by Node 2. As shown on the right side, once Node 1 moves into radiorange, Node 2’s cluster will be disbanded and replaced by three clusters.This reorganization is unnecessary, since Node 1 could have simply joinedthe existing cluster.
Figure 4: Instability of Lowest ID Clustering. As shown on the left, Node1 is moving into range of Node 2’s cluster. Once Node 1 is in radio range,Node 2’s cluster will be reorganized.
In order to address this instability, the clustering algorithm introducedby the Clusterhead-Gateway Switch Routing Protocol preserves clusterheadelections under all but two conditions [27]. The set of clusterheads changesonly when two clusterheads come into contact with each other, or when anode loses contact with all clusterheads.
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4.2 Greedy CDS Construction
Das et al. [35, 33, 66] propose the distributed implementations of the twogreedy algorithms given by Guha and Khuller in [44].
The first algorithm grows a CDS from one node with maximum de-gree. Thus the first step involves selecting the node with the highest degree.Therefore a node must know the degree of all nodes in the graph. On theother hand, each iterative step selects either a one- or two-edged path em-anating from the current CDS, thus the nodes in the CDS must know thenumber of unmarked neighbors for all nodes one and two hops from theCDS. These two requirements force the flooding of degree information inthe network. This algorithm generates a CDS with approximation ratio of2H(∆) in O(|C|(∆ + |C|)) time, using the O(n|C|) messages, where theharmonic function H(∆) =
∑∆i=1 1/i ≤ ln(∆) + 1, n is the total number of
vertices, and C represents the generated CDS.
The second algorithm first computes a dominating set and then selectsadditional nodes to connect the set. In order to locally select nodes for thedominating set in the first stage, an unmarked node compares its effectivedegree, the number of unmarked neighbors, with the effective degrees of allits neighbors in two-hop neighborhood. The greedy algorithm iterativelyadds the node with maximum effective degree to the dominating set. Thefirst stage terminates when a dominating set is achieved. The second stageconnects the obtained components using a distributed minimum spanningtree algorithm, with the goal of adding as few nodes as possible. To do this,each edge is assigned a weight equal to the number of endpoints not in thedominating set. At the end, the interior nodes in the resulting spanning treecompose a connected dominating set. This algorithm has time complexityof O((n+ |C|)∆), and message complexity of O(n|C|+m+n log(n)). It ap-proximates the MCDS with a ratio of 2H(∆)+1, where m is the cardinalityof the edge set.
Das et al. [35, 33, 66] handle CDS maintenance differently in the caseof single-node movement versus multiple-node movement. If a single nodemoves, the CDS can be updated locally. When more than one node moves,the moves can be treated as many single-node moves if they have no over-lapping neighborhoods. However, an entirely new CDS computation maybe needed in the case of overlapping neighborhoods.
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4.3 MIS Based CDS Construction
Algorithms in this category compute and connect an MIS. But how an MISis computed and connected differs from algorithm to algorithm. One cancompute an MIS based on either single leader [2, 3, 4, 5, 13, 16, 23, 69] ormultiple leaders [6, 25]. The MIS can be connected after the constructionis over [2, 3, 4, 5, 6, 13, 16, 23, 25, 69], or one can compute and connect anMIS simultaneously [23, 25]. Note that single leader based MIS constructionneeds a leader-election algorithm, which takes O(n log n) messages [10, 28].Nodes with maximum degree or id among all neighbors can serve as leaders inmultiple leader based MIS construction, thus the corresponding algorithmshave lower message complexity [6, 25].
Note that the algorithm provided by [58] also relies on an MIS. But wewill address it in detail in Subsection 4.6, as the major contribution in [58]is the exploitation of a Steiner tree with minimum number of Steiner pointsto connect an MIS.
4.3.1 Alzoubi and Wan’s Single Leader Algorithm
Alzoubi and Wan’s algorithms [4, 5, 69] utilize the properties of unit-diskgraphs (UDGs) to prove their performance bounds. By definition, an MISis adjacent to every node in the graph. Due to the geographic constraintsimposed by a UDG, a node is adjacent to at most five independent neighbors[55]. Therefore, an arbitrary MIS can contain no more than five times thenumber of nodes in the minimum-sized MIS. This observation forms thebasis of the proof of the performance bounds for all algorithms in [4, 5, 69].
Alzoubi et al. [4, 5, 69] provide two versions of an algorithm to con-struct the dominating set for a wireless network. In both algorithms, theyfirst employ the distributed leader election algorithm [28] to construct arooted spanning tree from the original network topology. Then, an iterativelabelling strategy is used to classify the nodes in the tree to be either black(dominator) or gray (dominatee), based on their ranks. The rank of a nodeis the ordered pair of its level (number of hops to the root of the spanningtree) and its ID.
The labelling process begins from the root node and finishes at the leaves.The node with the lowest rank marks itself black and broadcasts a DOM-INATOR message. The marking process then continues according to thefollowing rules:
• If the first message that a node receives is a DOMINATOR message,it marks itself gray and broadcasts a DOMINATEE message.
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• If a node received DOMINATEE messages from all its lower rankneighbors, it marks itself black and sends a dominator message.
The marking process finishes when it reaches the leaf nodes. At that time,the set of black nodes form an MIS, incorporating alternate levels of thespanning tree. Since the nodes are on alternating levels of the spanningtree, the distance between any subset of the MIS and its complement isexactly two hops away.
The final phase connects the nodes in the MIS to form a CDS, usingINVITE and JOIN messages. Initially, the root joins the CDS and broad-casts an INVITE message. The INVITE message is relayed to all two-hopneighbors out of the current CDS. When a black node receives the INVITEmessage for the first time, it joins the dominating tree together with the graynode, which relayed the message. It then initiates an INVITE message. Theprocess terminates when all the black nodes join the CDS.
This algorithm has time complexity of O(n), and message complexity ofO(n log(n)). The resulting CDS has a size of at most 8opt + 1.
0
7 1
11 9 6
3
10 8 2
12 4
5
0
7 1
11 9 6
3
10 8 2
12 4
5
0
7 1
11 9 6
3
10 8 2
12 4
5
(a) (b) (c)
Figure 5: An example of Alzoubi and Wan’s single leader algorithms forCDS construction.
Figure 5 illustrates the action of these algorithms. In this graph, node 0 isthe root of the spanning tree that is constructed by using the leader electionalgorithm. The solid lines represent the edges of the rooted spanning tree,and the dashed lines represent other edges in the UDG. Node 0 is markedblack first and broadcasts a DOMINATOR message (solid arrows in Figure5(a)). After receiving this message, nodes 2, 4, and 12 are marked gray andbroadcast DOMINATEE messages. (For simplicity, only the DOMINATEEmessages from node 4 are shown as the dashed arrows in Figure 5(a)). Thennode 5 is selected to be a DOMINATOR as it has received DOMINATEEmessages from all its lower rank neighbors (node 4 only). Figure 5(b) showsthe colors of the nodes when the labelling process finishes. The final process
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builds the dominating tree from the root. The INVITE message (solid arrowin Figure 5(b)) is sent from node 0, and it is relayed to its two-hop blackneighbors 3, 5 and 7. These black nodes join the dominating tree, as well astheir relaying gray nodes 2 and 4. The thick links in Figure 5(b) illustrate theedges in the final dominating tree. All the nodes in the tree form a connecteddominating set. The improved approach in [5] merges the MIS constructionwith the dominating tree building processes. As shown in Figure 5(c), node2 is colored black when it first receives a DOMINATOR message from itschild 3, and node 4 is marked black for the same reason. Finally, all theblack nodes form a connected dominating set.
4.3.2 Cheng’s Single Leader Algorithm
Cheng et al. [13, 16, 25, 23] present two algorithms for growing a connecteddominating set from a leader node. Compared with the work of Alzoubi etal. [4, 5, 69], they introduce a new active state for vertices to describe thecurrent labelling set of vertex nodes. With the help of this new concept,either cost-aware or degree-aware optimization can be achieved.
Their first algorithm is cost-aware. Each host has a local cost, whichserves as the selection criterion together with its ID. At the beginning, allvertices are in initial state with white color. The leader starts the algorithmby marking itself black and becoming a dominator. A white node goes tobe a dominatee (gray) if one of its neighbors becomes a dominator. A non-active white vertex changes to status active if one of its neighbors becomesa dominatee. Its color still keeps white. Then, an active node with thesmallest cost among all its active neighbors will compete to be a dominator.Its minimum cost gray parent also changes to serve as its dominator (black),ensuring the connectivity of the dominating tree. Finally, all black leaf nodescan change back to be dominatees (gray). This process terminates when allnodes are colored gray or black, and all the black nodes form a connecteddominating set.
This algorithm has the time complexity of O(n), and the message com-plexity of O(n log n), which is dominated by leader election. The perfor-mance ratio is 8opt + 1, the same as that of the algorithm in [4].
An example that illustrates the application of Cheng’s single leader al-gorithm is shown in Figure 6. There are 9 hosts and 12 links. We assumehost IDs are the costs. Host 0 is the leader. In the beginning, node 0 iscolored black, serving as a dominator. Nodes 1 and 5 are then colored gray.Nodes 2 and 6 become active. Their competition results in the winner ofnode 2. Node 2 colors itself black and invites node 1 to be its dominator.
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0
8
7
6 5
4 3
2
1
Figure 6: An example of unit-disk graph G containing 9 hosts and 12 links.
0
8
7
6 5
4 3
2
1
Figure 7: The computed connected dominating set from Cheng’s singleleader algorithm contains hosts 0, 1, 2, 3, 4, 7. The optimal solution con-tains 1, 2, 3, 4, 7
Therefore node 3 is colored gray and node 4 becomes active. This processcontinues until no white nodes left. The result is demonstrated in Figure 7.All black nodes form a CDS.
Note that in this algorithm, a dominating tree is grown from the leader.The resultant CDS contains two subsets: one changes color from white toblack directly, and the other colors themselves from white to gray thento black. If the determination of the second subset is delayed until thefirst subset is constructed, and the criterion for changing color from grayto black is based on the number of black neighbors a gray node has, thedegree-based algorithm described in [23] is obtained. This algorithm hasperformance ratio 8. However, since effective degree information (numberof white neighbors) needs to be updated during the algorithm execution,the degree-aware algorithm takes higher number of messages compared tothe cost-aware algorithm, even though their time and message complexitiesare the same. Min and Du [57] extends the second algorithm to considerreliable virtual backbone construction in MANET.
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4.3.3 Alzoubi’s Multiple Leader algorithm
Single leader based CDS construction has message complexity Ω(n log n),which is dominated by leader election. In single leader CDS construction, thedependence on a rooted tree renders the maintenance of the CDS extremelydifficult. To decrease the protocol overhead caused by the large volumeof message exchange, and to improve the structure of the computed CDS,multiple leader based algorithms [6, 25] are proposed. Compared with singleleader based CDS construction algorithms, the maintenance of the CDSconstructed based on multiple leaders may be faster, as it does not rely ona spanning tree. In this subsection, we will study the algorithm introducedby Alzoubi et al. [6]. In next subsection, the algorithm provided by Chenget al. [25] will be elaborated.
The algorithm proposed by Alzoubi et al. [6] constructs a CDS in a UDGwith size at most 192 · |OPT |+ 48. The message complexity is O(n). Thisalgorithm does not use a rooted spanning tree. Initially all the nodes arecandidates. Whenever the ID of a node becomes the smallest among all of itsone-hop neighbors, it will change its status to dominator. Consequently, itscandidate neighbors become dominatees. After all nodes change status, eachdominator node identifies a path of at most three hops to another dominatorwith larger ID. The candidate nodes on this path become connectors. Alldominators and connectors compose a connected dominating set.
Figure 8: An example of Message-Optimal CDS Construction [6].
The example in [6] can illustrate the execution of this algorithm. Asshown in Figure 8(b), nodes 1, 2, 3 and 4 declare themselves to be domi-
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nators in the beginning. Then node 5 declares itself as a connector on thepaths from 1/2 to 3/4, so do nodes 6 and 7. Finally all the connectors anddominators form a connected dominating set.
4.3.4 Cheng’s Multiple Leader Algorithm
Cheng’s multiple leader algorithm [25] is motivated by [6]. Initially eachnode with smallest ID among its one-hop neighbors becomes a leader. Aforest with each tree rooted at a leader is constructed first. Then a chainof one or two nodes are computed to connect neighboring trees. This al-gorithm has linear time and message complexities, and generates a CDSwith size at most 147 · |OPT |+ 33. The performance analysis explores theneighboring property of nodes in a tree, thus the algorithm achieves betterperformance ratio compared to the multiple leader algorithm in [6]. Thisalgorithm achieves linear time and message complexities.
4.3.5 Single Leader vs. Multiple Leader
Although single leader based CDS construction algorithms provide betterperformance bounds, their non-localized construction may render them un-usable. Since each algorithm has a time complexity of O(n), nodes maymove during CDS construction such that the result of the algorithm is nota CDS. Thus, the ultimate goal of the CDS construction is to develop trulylocalized algorithms that have constant time complexity. Since the worst-case time complexity of the two multiple leader based algorithms [6, 25] isO(n) due to the MIS or forest construction, these algorithms are not purelylocalized in a strict sense.
Nonetheless, multiple leader based algorithms in [6, 25] do representprogress in message complexity - they achieve the optimal message com-plexity of O(n). Message complexity of a CDS construction algorithm con-tributes to the protocol overhead. It also plays an important role in theeffectiveness and efficiency analysis of the protocol. Single leader basedCDS construction algorithms [4, 5, 69, 23, 16] have message complexityO(n log n). The direct distributed implementation of Guha and Khuller’salgorithms have message complexities O(n|C|) and O(n|C| + m + nlog(n))[35, 33, 66]. These algorithms have better performance ratio. Therefore asa trade-off the performance ratios of [6] and [25] are much higher.
The two truly localized algorithms are presented by Wu and Li in [76],and by Adjih, Jacquet, and Viennot in [1], which will be discussed in Subsec-tions 4.4 and 4.5. Both algorithms need two-hop neighborhood information.
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Wu and Li’s algorithm first creates a larger CDS by selecting more nodesthan needed, then prunes the set of selected nodes to get a CDS with smallersize. The algorithm proposed by Adjih, Jacquet, and Viennot relies on amultipoint relaying set.
4.4 Pruning Based CDS Construction
There are two pruning-based CDS construction algorithms [14, 76] proposedin the context of ad hoc and sensor networks. We will study them in thefollowing subsections.
4.4.1 Wu and Li’s Algorithm
Wu et al.’s work [76, 31] proposes a completely localized algorithm to con-struct CDS in general graphs. Initially all vertices are unmarked. Theyexchange their open neighborhood information with their one-hop neigh-bors. Therefore each node knows all of its two-hop neighbors. The markingprocess uses the following simple rule: any vertex having two unconnectedneighbors is marked as a dominator. The set of marked vertices form aconnected dominating set, with a lot of redundant nodes. Two pruningprinciples are provided to post-process the dominating set, based on theneighborhood subset coverage. A node u can be taken out from S, theCDS, if there exists a node v with higher ID such that the closed neighborset of u is a subset of the closed neighbor set of v. For the same reason,a node u will be deleted from S when two of its connected neighbors in Swith higher IDs can cover all of u’s neighbors. This pruning idea is gener-ated to the following general rule [32]: a node u can be removed from S ifthere exist k connected neighbors with higher IDs in S that can cover all u’sneighbors. Wu et al. extend their work to calculate power-aware connecteddominating sets [72, 77], by considering the power property for all the nodesas a criterion for the post pruning.
This idea is also extended to directed graphs. Due to differences intransmission ranges, or the hidden terminal problem [68] in wireless net-works, some links in an ad hoc network may be unidirectional. In order toapply this algorithm to a directed graph model, neighboring vertices of acertain node are classified into a dominating neighbor set and an absorbentneighbor set in terms of the directions of the connected edges [70]. Figure10 illustrates the dominating and absorbent neighbor sets of vertex u. Inthis case, the objective is to find a small set that is both dominating and ab-sorbent for a given directed graph. The original marking process is adapted
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Figure 9: Examples of two pruning principles to eliminate redundant nodes[72].
as follows: a node u is added into the dominating and absorbent set, whenthere exists a node w in its dominating set and another node v in its ab-sorbent set which can only be connected via u. Then similar post-processprinciples are used to delete the redundant nodes in the resulted dominatingand absorbent set.
Figure 10: Dominating (absorbent) neighbor set of vertex u [70].
The performance ratio for Wu’s algorithm is proved to be O(n)[69]. Forthe post-processing, the nodes need to know their two hop neighbors. Thusthe algorithm has time complexity of O(∆3) and message complexity ofΘ(m), which indicate that the maintenance of the CDS constructed by Wu’salgorithm is relatively easier. In fact one major advantage of Wu and Li’salgorithm is its locality of maintenance. Mobile hosts may switch off oron at any time for power efficiency. In addition to this switching, CDSmaintenance may be required due to mobile hosts’ movements. Comparedto other CDS algorithms, Wu’s algorithm only requires neighbors of a nodeto update their status when it changes switching status or location.
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4.4.2 Butenko’s algorithm
The centralized version of this algorithm [14, 15] is discussed in Subsection3.5. Its distributed implementation has higher message complexity becauseglobal connectivity needs to be checked when examining each node. Thiscan be done through any distributed spanning tree algorithm [9].
The algorithm starts from the node with minimum degree, which canbe found by modified leader-election algorithms in [28]. Let u be the nodeunder consideration at the current step. If removing u causes the CDSdisconnected, then we color u black. u then selects its non-black neighborwith minimum effective degree (the number of non-black neighbors withinthe set) for consideration in next step. If it is OK to remove u, u willselect a neighbor with minimum effective degree, if u does not have a blackneighbor, for next step. If u does have a black neighbor v, then v will chooseits neighbor with minimum effective degree for next step. This procedurewill be continued until all nodes have been examined.
4.5 Multipoint Relaying Based CDS Construction
Multipoint relaying is a technique to allow each node u select a minimumforwarding set [61, 51, 17] from N(u) to cover N(N(u)). Finding a multipointrelay set (MRS) with minimum size is NP-Complete [61]. Refs. [61] and [51]independently design different log n factor greedy heuristics where n is thetotal number of hosts. Ref. [17] provides a sophisticated approximationalgorithm with constant performance ratio 6. Multipoint relaying is mainlyused for flooding control to decrease protocol overhead [46]. Recently Adjih,Jacquet, and Viennot [1] propose a localized heuristic to generate a CDSbased on multipoint relaying. Their idea is sketched below.
Each node first compute a multipoint relay set, a subset of one-hop neigh-bors that can cover all the two-hop neighbors. The a CDS is constructed bytwo rules. The first rule puts all nodes with smallest ID among their neigh-bors into the CDS; The second rule puts a node into the CDS if the nodeis a member of the MRS of its smallest ID neighbor. Wu [71] enhanced thefirst rule by selecting the node that has at least two disconnected neighborsand has the smallest ID among all its neighbors. Chen and Shen [22] claimsthat by considering node degree instead of ID, the constructed CDS mayhave smaller size. The correctness of this heuristic is proved in [1] but noperformance analysis is available.
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4.6 Steiner Tree Based CDS Construction
The second algorithm proposed in Guha and Khuller [44] is Steiner treebased. Its distributed implementation [35, 33, 34] applies a spanning treeto approximate the computation of Steiner points. The single leader basedalgorithms proposed in [2, 3, 4, 5, 16, 23, 69] also compute Steiner pointsto connect the MIS. These algorithms have been discussed in Subsections4.2 and 4.3, thus they will not be repeated here. In this subsection, we willstudy how the centralized algorithm proposed in [58] is implemented in adistributed fashion.
As mentioned by Subsection 3.4, Min’s CDS construction algorithm ap-plies Steiner tree with minimum number of Steiner points to connect anMIS with the following property: any subset of the MIS has hop distancetwo with its compliment. We need to consider two phases in the distributedimplementation: the construction of the MIS and the computation of theSteiner points.
Cheng et al. [25] and Wan et al. [69] both imply heuristics to computean MIS in a distributed fashion. Here we rephrase the simple idea basedon [25]. Assume all nodes are white initially. In the first step, color thenode with smallest id black. Color all its one-hop neighbors gray; color itstwo-hop neighbors yellow. At each step, choose a yellow node with small-est id among all its yellow neighbors and color it black. Color its one-hopwhite/yellow neighbors gray; color its two-hop white neighbors yellow. Re-peat this procedure until all nodes are colored either black or gray. All blacknodes form an MIS.
The distributed computation of the Steiner tree with minimum numberof Steiner points is non-trivial, as demonstrated by [58]. The idea is statedbelow: each gray node u (a node not in the MIS) keeps track of the neigh-boring connected black components. Its competitor set includes all its grayneighbors and all the gray nodes adjacent to its neighboring connected blackcomponents. u becomes a Steiner point if and only if it is adjacent to themaximum number of black components among all its competitors. It is clearthat this distributed implementation has very high message complexity.
4.7 Proactively Considering Nodal Mobility in CDS Con-struction
Most of the previous algorithms have attempted to address nodal move-ments by incorporating a maintenance phase in which the CDS can be re-constructed when nodes move. However, in the case of high nodal mobility,
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this reactive approach may not yield a stable infrastructure. A numberof DS algorithms have been developed that attempt to proactively utilizemobility information in an attempt to create stable clusters.
The Mobility Adaptive Clustering Algorithm, for example, selects slowmoving nodes for inclusion in a WCDS [11]. When nodes move randomly,a fast moving clusterhead is likely to encounter another clusterhead soonerthan a slow moving one. Furthermore, the open neighbor sets of fast movingnodes will exhibit more change than those of slow moving nodes. Therefore,this algorithm selects slow moving nodes, which are more likely to havestable links.
Mobility-Based Clustering also considers nodal movement in the creationof a WCDS [8]. This WCDS approach has three notable features:
• It uses relative mobility information to create clusters.
• It allows cluster members to be L hops from the clusterhead, whereL ≥ 1, in order to increase the stability of clusters.
• Rather than seeking a minimal WCDS, it allows for clusterheads tocome into contact with each other if the clusterheads are going indifferent directions.
In this WCDS protocol, neighbors periodically exchange their position andvelocity information with their neighbors. A node calculates the relativevelocity and the relative mobility between itself and all of its neighbors.Relative mobility is an average of the magnitude of the relative velocity vec-tor in the recent past. Once relative mobility has been measured, there isan initial cluster creation where, a node elects as its clusterhead its neighborwith the lowest id, whose relative mobility falls below a user-defined thresh-old. As this step is performed, a clusterhead which has been elected, mayjoin another cluster, with the two clusters then merging into one, subject tothe L-hop rule. Cluster maintenance is done sparingly, only when a nodeloses contact with its clusterhead, and encounters a node whose relativemobility is less than the threshold.
(α,t)-Clusters takes a similar approach in the creation of a WCDS. Itrequires that nodes in a cluster be mutually reachable after time t with aprobability of α [56]. In their simulations, nodes move in a random walk-based mobility. They derive the probability that two nodes are mutuallyreachable based on this mobility pattern, and current mobility information.
The Clustering for Open Inter-vehicle communication Networks (COIN)algorithm arises because clustering protocols designed random mobility pat-terns perform poorly for inter-vehicle communication [12]. This algorithm
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was tested with nodal mobility generated from microscopic vehicular trafficsimulators. Vehicular mobility is highly constrained by the layout of roadnetwork, by traffic control devices, and by surrounding vehicles. Vehiclemovement is characterized by high rates of speed, producing very high rela-tive velocities. Driver behavior has a significant impact of mobility patternsboth in the near term and in the long term. In the near term, vehicle move-ments vary dramatically based on individual lane changing, braking, andpassing behaviors. In the long term, mobility is affected by the variationsin the intended destination of a driver. The COIN algorithm enhances theprediction of future mobility by incorporating driver intentions into the pre-diction algorithm. The destination of a driver could be gleaned either froman on-board route guidance system, or through a statistical analysis of pre-vious trips on the current roadway. The paper also identifies a source ofinstability in clustering - oscillatory inter-vehicle distances between vehicleswith low relative mobility. Examples of this phenomenon can be found instop and go traffic, as vehicles pass through four-way stop signs, or as vehi-cles slow to navigate curves in the roadway. In order to accommodate thisphenomenon the algorithm relaxes the condition that no two clusterheadswith low relative mobility be within radio contact. Instead, a design param-eter, the inter-cluster minimum distance, specifies the minimum distancebetween two clusterheads.
5 Analysis of Distributed CDS Algorithms
The algorithms presented in the previous section represent an evolution ofdistributed CDS algorithms designed for use in a variety of wireless appli-cations. This evolution has been driven by both improvements in efficiencyand stability and by the design requirements of the consumer applications.The following analysis presents the evolution at these two levels. First, wedescribe the development of a toolbox of techniques for CDS construction.Then, at a higher level, we elicit the manner in which the needs of theconsumer applications dictate the selection of specific techniques from thistoolbox.
Since non-localized algorithms rely on global information, their timecomplexity is at least O(n) as in [33]. Purely localized CDS constructionalgorithms operate much quicker, with a complexity related to the maximaldegree, typically O(∆3) [1, 76]. This quicker execution time comes at a costof a larger CDS.
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5.1 A Toolbox for CDS Construction
5.1.1 WCDS Construction Techniques
WCDS construction plays an important role not only in clustering algo-rithms, but also in a number of CDS construction algorithms, which beginwith the selection of a WCDS. The first WCDS construction algorithm se-lected the lowest ID neighbor as clusterhead [40]. In newer algorithms,techniques have been developed to reduce the size of the clusterhead set,improve the stability of the clusterhead set, and promote fairness in cluster-head selection.
The size of the WCDS can be reduced by choosing nodes with the highestdegree as clusterheads [43]. Although this will reduce the size of the WCDS,if a network with mobile nodes, this may increase WCDS instability sincenodal degree varies while ID does not.
Since stability of the WCDS is a key factor in many applications, clus-terhead stability has been improved through preservation of clusterheadelections and multi-hop clusters. Clusterhead selection need not changewhenever the topology of the network changes; rather, reelections mightonly occur when clusterheads move into range of each other or when a nodemoves out of range of all clusterheads [27]. Multi-hop clusters can improvecluster stability by increasing the area covered by a single clusterhead [7].
Fairness may also be a problem with Lowest ID clustering, since the bur-dens of being a clusterhead are primarily borne by those nodes with lowerID’s. However, mitigating this shortcoming should not be done in a waythat decreases stability, as in highest connectivity clustering without clus-terhead election preservation. For example, Min-Max D-Clusters attemptedto address this issue by selecting both low and high id nodes as clusterheads[7].
5.1.2 CDS Construction Techniques
We have discussed example algorithms exploiting the following distributedCDS construction techniques: greedy, MIS based, Steiner tree based, prun-ing based, and multipoint relaying. We briefly discuss them in the following.
Das et al.’s greedy algorithms [35, 33, 66] are the distributed implemen-tations of Guha and Khuller’s algorithms in [44]. These heuristics have highmessage complexity due to the global selection of the greedy choice. Theyare the first distributed ones proposed for MCDS computation in the con-text of wireless ad hoc networks. One feature of their scheme is to store theglobal topology information only in dominating nodes. This reduces access
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and update overheads for routing. However, their CDS construction requirestwo hop neighborhood knowledge. The generated CDS has high approxima-tion ratio and high implementation complexities (in message and time). Inaddition, it is not clear in their algorithm description how each individualnode is informed on when to start the second stage. The CDS maintenanceis expensive too, as their approaches need to maintain a spanning tree.
The MIS based algorithms [6, 25, 23, 69] compute and connect an MIS.Their performance analysis in unit-disk graphs take the advantage of therelationship between an MIS and OPT. Algorithms in this category usuallyhave good performance bound and time/message complexities. They onlyneed one-hop neighborhood information. However, the single leader basedalgorithms [13, 16, 25, 23] require leader election. This drawback makesthem difficult to support localized CDS maintenance. Multiple leader basedalgorithms [6, 25] are optimal in message complexity. Compared to singleleader algorithms, they are relatively more practical in local maintenancesince they obviates the rooted spanning tree construction.
Pruning based algorithms [14, 76] prune a large CDS. Wu and Li’s algo-rithm [76] is the first purely localized CDS construction heuristic. Butenkoet al.’s algorithm [14] has high message complexity due to the global con-nectivity checking.
Wu and Li’s algorithm [70, 76, 72] is very simple. The localized propertymakes the CDS maintenance easier. However, there is no performance anal-ysis in the original paper [76, 72], which incorrectly analyzes the algorithm’stime complexity. Ref. [4] corrects the mistake in [76], and proves that Wuand Li’s algorithm has a linear performance ratio. This algorithm needs atleast two-hop neighborhood information. It is presented based on the gen-eral graph model, and is extended to directed graph [70]. This is importantfor wireless network, as either the disparity of transmission range or the hid-den terminal problem in physical wireless networks can cause unidirectionallinks.
The Steiner tree based CDS construction heuristic in [58] uses a Steinertree with minimum number of Steiner points to connect a dominating set,usually an MIS. Computing Steiner points requires large number of messageexchange.
The multipoint relaying based heuristic [1] is pure localized. This algo-rithm selects CDS from a multipoint relay set. No complexity analysis forthis algorithm in literature.
We summarize the above analysis for major algorithms in the followingtable.
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5.2 Selection of CDS Construction Methods
The design goals of the application dictate the selection from the differ-ent techniques for creating a virtual network infrastructure. Depending onthe needs of the application, either a WCDS or CDS will be appropriate.The application requirements will also dictate the balance between boundedperformance and fast operation. Finally, requirements for stability will af-fect the size and characteristics of the dominating set, as well as the nodeselection methodology.
The selection of a WCDS or a CDS depends on the communication re-quirements between nodes within the dominating set. Routing applicationstend to rely on CDS, since messages must be forwarded along a backbone.However, if the network topology is highly unstable, a WCDS may be a bet-ter choice since its smaller size makes it easier to maintain. Likewise, whileintra-cluster coordination functions might be managed within a WCDS,inter-cluster coordination is probably more easily handled by a CDS. Soif adjacent clusters are using orthogonal codes or different frequency bands,a WCDS could manage media access within each cluster. However, for asystem-wide media access coordination, a CDS may be more efficient sinceit includes nodes needed for clusterhead communication.
In addition to selecting the underlying virtual network infrastructure,the application developer must select the appropriate balance between per-formance bounds and fast operation. As seen in the previous examples,significantly better performance bounds are achievable if the dominating setis constructed serially as opposed to in parallel. However, the applicationmay not be able to accommodate the additional time required to constructthe dominating set. Moreover, with mobile nodes, then by the time thedominating set is constructed, the network topology may have changed suchthat the result of the algorithm is not a dominating set.
In addition to performance considerations, the stability requirementsof the dominating set are also crucial for applications that require long-lasting dominating sets for mobile nodes. The stability of dominating setcan be measured either from the perspective of a dominating set or fromthe perspective of a node outside of the dominating set. One may seek toreduce the rate of CDS change. Alternatively, one may seek to increase theduration that a node is associated with a given node in the dominating set.An example of the latter would be to increase the time that a node is in thecluster of a given clusterhead.
Depending on the definition of stability, different techniques can be em-ployed. If one seeks to promote the stability of the CDS, then the selection
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of low mobility nodes as members of the CDS is suitable approach.However, if one also seeks to promote the association stability between
nodes and members of the CDS, then one must create these associationsbased on the predictions of relative mobility between nodes. Furthermore,these predictions are domain-specific - predictions that reflect random mo-bility are likely to be unsuitable for domains where mobility is constrained.
A common technique in many of these algorithms is enlarging the sizeof the dominating set. This achieves greater stability by relaxing the con-straints on the number of elements in a dominating set that are within radiorange of each other. For the vehicular mobility case, for example, one endsup with at least two dominating sets - one for each direction of vehicle traffic.The overlap in these sets allows for both stability of the CDS, and stabilityof node association with clusterheads.
6 Conclusions and Discussions
Dominating sets have proven to be an effective construct within which tosolve a variety of problems that arise in wireless networks. Applicationsthat use dominating sets include media access coordination, unicast andmulticast routing, and energy efficiency.
The needs of these consumer applications drive the design goals of thedominating set construction algorithms. In our review of these algorithms,we have examined a number of these considerations. The most obvious isthe choice of target set - a WCDS, CDS, or a d-hop dominating set. Further-more, the large differences in time and message complexity are the result oftradeoffs between fast operation and bandwidth efficiency versus dominatingset size. In addition, these algorithms may need to address nodal mobility,if the dominating set is not transitory. Some algorithms have attempted toaddress this problem through the creation of localized maintenance routinesthat seek to avoid CDS recreation through a rerunning of the constructionalgorithm. Others have also included proactive consideration of nodal mo-bility in order to promote CDS stability.
As ad hoc wireless networking moves from the research labs to the realworld, additional requirements will certainly arise. Issues that will likelyarise include the security and survivability of CDS-based applications. Ininfrastructure-based wireless networks, the infrastructure plays a key role forsecurity purposes. A perimeter is established around the infrastructure, reg-ulating access to the network. A CDS could perform similar actions throughforming a virtual network infrastructure. However, significant issues must
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first be addressed including the real-time assurance of the trustworthinessof mobile hosts. Similarly, performance of the DS algorithms in the face ofDenial of Service and other security attacks must be evaluated in order toensure robust, survivable network functionality.
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